Let G be an undirected graph with m edges and n vertices. A spanner of G is a subgraph which preserves approximate distances between all pairs of vertices. An f-vertex fault-tolerant spanner is a subgraph which preserves approximate distances, under the failure of any set of at most f vertices. The contribution of this paper is twofold: we present algorithms for computing fault-tolerant spanners, and propose streaming algorithms for computing spanners in very small internal memory. In particular, we give algorithms for computing f-vertex fault-tolerant (3,2)- and (2,1)-spanners of G with the following bounds: our (3,2)-spanner contains O(f4/3n4/3) edges and can be computed in time Õ(f2m), while our(2, 1)-spanner contains O(fn3/2) edges and can be computed in time [Formula: see text]. Both algorithms improve significantly on previously known bounds. Assume that the graph G is presented as an input stream of edges, which may appear in any arbitrary order, and that we do not know in advance m and n. We show how to compute efficiently (3, 2)- and (2, 1)-spanners of G, using only very small internal memory and as low access external memory device. Our spanners have asymptotically optimal size and the I/O complexity of our algorithms for computing such spanners is optimal upto apolylogarithmic factor. Our f-vertex fault-tolerant (3, 2)- and (2, 1)-spanners can also be computed efficiently in the same computational model described above.
On Resilient Graph Spanners